Fractional relaxation and fractional oscillation models involving Erdelyi-Kober integrals

TL;DR

This paper explores fractional relaxation and oscillation models using Erdelyi-Kober integrals, providing analytical and numerical solutions, and advancing methods for fractional damped oscillators with time-varying coefficients.

Contribution

It introduces generalized fractional models with Erdelyi-Kober integrals and develops numerical and analytical solutions, including new approaches for fractional oscillators with time-varying coefficients.

Findings

Solutions expressed in Saigo-Kilbas Mittag-Leffler functions

Numerical methods adapted from existing approaches

New insights into fractional damped oscillators with variable coefficients

Abstract

We consider fractional relaxation and fractional oscillation equations involving Erdelyi-Kober integrals. In terms of Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing Riemann-Liouville integrals with Erdelyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.